EB-GLS: an improved guided local search based on the big valley structure

Abstract

Local search is a basic building block in memetic algorithms. Guided local search (GLS) can improve the efficiency of local search. By changing the guide function, GLS guides a local search to escape from locally optimal solutions and find better solutions. The key component of GLS is its penalizing mechanism which determines which feature is selected to penalize when the search is trapped in a locally optimal solution. The original GLS penalizing mechanism only makes use of the cost and the current penalty value of each feature. It is well known that many combinatorial optimization problems have a big valley structure, i.e., the better a solution is, the more the chance it is closer to a globally optimal solution. This paper proposes to use big valley structure assumption to improve the GLS penalizing mechanism. An improved GLS algorithm called elite biased GLS (EB-GLS) is proposed. EB-GLS records and maintains an elite solution as an estimate of the globally optimal solutions, and reduces the chance of penalizing the features in this solution. We have systematically tested the proposed algorithm on the symmetric traveling salesman problem. Experimental results show that EB-GLS is significantly better than GLS.

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Notes

Acknowledgements

The work described in this paper was supported by a grant from ANR/RCC Joint Research Scheme sponsored by the Research Grants Council of the Hong Kong Special Administrative Region, China and France National Research Agency (Project No. A-CityU101/16).

Appendix

In Sect. 4, we state that att532 has the big valley structure, while u2319 does not have the big valley structure. Our statements are based on the landscape sampling experiment conducted on these two instances, in which 1000 runs of GLS and 1000 runs of EB-GLS are executed until finding the global optimum. During each run, the best solutions found so far are recorded and the final global optimum is also recorded. In fact, we conduct the same landscape sampling experiment on another eight instances. By analyzing the results, we conclude that all these eight instances satisfy the requirements of the big valley structure we defined in Sect. 4. Table 5 shows the landscape sampling results on these eight instances. Figure 13 shows the scatter plots of the recorded best solutions found so far.

Table 5

The landscape sampling results on eight selected TSPLIB instances

Instances

City num.

\(N_{opt}\)

\(D_{o,min}\)

\(D_{o,avr}\)

\(D_{o,max}\)

FDC

rd400

400

8

3

21

36

0.83

gr431

431

2

13

13

13

0.76

pcb442

442

2000

6

40

69

0.79

pa561

561

2000

6

44

82

0.83

u574

574

4

2

4

6

0.84

rat575

575

2

3

3

3

0.85

rat783

783

811

2

15

31

0.89

u1432

1432

2000

191

269

342

0.82

\(N_{opt}\) is the number of the unique global optima found by these 2000 runs. \(D_{o,min}\) (\(D_{o,avr}\),\(D_{o,max}\)) is the minimum (average, maximum) distance between the global optima. The last column is the FDC value of the recorded best solutions found so far during the 2000 runs

The scatter plots of the recorded best solutions found so far during 1000 runs of GLS and 1000 runs of EB-GLS on eight selected instances. The cost difference to the globally optimal cost (vertical axis) is plotted against the distance to the nearest globally optimal solution (horizontal axis). a rd400. b gr431. c pcb442. d pa561. e u574. (f) rat575. (g) rat783. h u1432

In Table 1 we present the comparison results between EB-GLS and GLS on the 33 TSPLIB instances with more than 1000 cities. Table 6 shows the comparison results on the other 76 TSPLIB instances with less than 1000 cities.

Table 6

Comparison results between EB-GLS and GLS on the TSPLIB instances with less than 1000 cities, the better metric values are marked by bold texts